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Taylor Polynomial Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A Taylor polynomial calculator computes the polynomial approximation of a differentiable function around a specified center point using Taylor's theorem. It calculates successive symbolic derivatives of the function, evaluates them at the center point, divides by the corresponding factorial to find the coefficients, and assembles the polynomial term-by-term. When centered at zero, the result is known as a Maclaurin polynomial.

Calculate the nth-degree Taylor polynomial (or Maclaurin series) of any differentiable function about a center point with step-by-step derivatives.

Quick Answer

Calculate the nth-degree Taylor or Maclaurin polynomial approximation of a function about a center point, with step-by-step symbolic derivatives.

Supports standard functions like exp(x), cos(x), sin(x), log(x), x^3 + 2x.
Setting a = 0 gives a Maclaurin series.
Degrees 0 through 10 are supported.
Compare f(x) vs. the Taylor polynomial approximation.
Taylor Approximation

Taylor Polynomial P4(x)

x - 0.166667x³

Functionsin(x)
Center point (a)0
Polynomial Degree (n)4
Exact Value f(1)0.841471
Approx. Value P(1)0.833333
Absolute Error0.00813765

Derivatives & Coefficients Table:

kf⁽ᵏ⁾(x)f⁽ᵏ⁾(a)cₖ = f⁽ᵏ⁾(a)/k!
0sin(x)00
1cos(x)11
2-sin(x)-0-0
3-cos(x)-1-0.166667
4sin(x)00
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How it works

What is Taylor's Theorem?

Taylor's theorem states that any function that is sufficiently differentiable near a point a can be approximated by a polynomial whose coefficients depend only on the derivatives of the function at that point.

The Taylor Polynomial Formula

The Taylor polynomial of degree n for a function f(x) centered at x = a is defined as:

Pₙ(x) = f(a) + f'(a)(x − a) + [f''(a)/2!](x − a)² + [f'''(a)/3!](x − a)³ + ... + [f⁽ⁿ⁾(a)/n!](x − a)ⁿ

Which can be written compactly in summation notation as:

Pₙ(x) = ∑ [ f⁽ᵏ⁾(a) / k! ] · (x − a)ᵏ   (from k = 0 to n)

Where:

  • f⁽ᵏ⁾(a) is the k-th derivative of f evaluated at x = a (with f⁽⁰⁾(a) = f(a)).
  • k! is the factorial of k (e.g., 3! = 3 × 2 × 1 = 6).
  • a is the center point of the approximation. When a = 0, the result is called the Maclaurin Polynomial.

Worked Example

Let's find the 2nd-degree Taylor polynomial for f(x) = ln(x) centered at a = 1:

  1. Evaluate f(a): f(1) = ln(1) = 0.
  2. Find and evaluate f'(x): f'(x) = 1/x, so f'(1) = 1. The term is 1·(x − 1).
  3. Find and evaluate f''(x): f''(x) = −1/x², so f''(1) = −1. The term is (−1 / 2!)·(x − 1)² = −0.5·(x − 1)².
  4. Combine: P₂(x) = 0 + 1(x − 1) − 0.5(x − 1)² = (x − 1) − 0.5(x − 1)².

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Frequently asked questions

A Taylor polynomial is a polynomial approximation of a function near a specific point. For a smooth function, the nth-degree Taylor polynomial has the same value and the same first n derivatives as the function at that center point. It is used to approximate transcendental functions like sin(x), cos(x), and e^x.

A Maclaurin polynomial is simply a Taylor polynomial centered at zero (a = 0). For example, the Maclaurin polynomial of degree 3 for sin(x) is x − x³/6.

Generally, as the degree (n) of the Taylor polynomial increases, the approximation becomes more accurate and covers a wider interval around the center point. However, outside of this radius of convergence, the polynomial may diverge rapidly from the actual function.

The coefficient cₖ for the (x − a)ᵏ term is calculated as the k-th derivative of the function evaluated at the center point (a), divided by k factorial: cₖ = f⁽ᵏ⁾(a) / k!.

The difference between the actual function and its Taylor polynomial is the remainder, Rₙ(x) = f(x) − Pₙ(x). According to Taylor's theorem, this error can be estimated using the Lagrange formula for the remainder, which depends on the (n+1)-th derivative of the function at some point between the center and the evaluation point.